On indecomposable $τ$-rigid modules over cluster-tilted algebras of tame type
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2017
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| _version_ | 1866911776591839232 |
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| author | Fu, Changjian Geng, Shengfei |
| author_facet | Fu, Changjian Geng, Shengfei |
| contents | For a given cluster-tilted algebra $A$ of tame type, it is proved that different indecomposable $τ$-rigid $A$-modules have different dimension vectors. This is motivated by Fomin-Zelevinsky's denominator conjecture for cluster algebras. As an application, we establish a weak version of the denominator conjecture for cluster algebras of tame type. Namely, we show that different cluster variables have different denominators with respect to a given cluster for a cluster algebra of tame type. Our approach involves Iyama-Yoshino's construction of subfactors of triangulated categories. In particular,we obtain a description of the subfactors of cluster categories of tame type with respect to an indecomposable rigid object, which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1705_10939 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On indecomposable $τ$-rigid modules over cluster-tilted algebras of tame type Fu, Changjian Geng, Shengfei Rings and Algebras Representation Theory For a given cluster-tilted algebra $A$ of tame type, it is proved that different indecomposable $τ$-rigid $A$-modules have different dimension vectors. This is motivated by Fomin-Zelevinsky's denominator conjecture for cluster algebras. As an application, we establish a weak version of the denominator conjecture for cluster algebras of tame type. Namely, we show that different cluster variables have different denominators with respect to a given cluster for a cluster algebra of tame type. Our approach involves Iyama-Yoshino's construction of subfactors of triangulated categories. In particular,we obtain a description of the subfactors of cluster categories of tame type with respect to an indecomposable rigid object, which is of independent interest. |
| title | On indecomposable $τ$-rigid modules over cluster-tilted algebras of tame type |
| topic | Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/1705.10939 |